Question

Find the ratio of $A:B:C$ such that given ratios \[A:B = 2:3\] and \[B:C = 4:7\].

Answer 1

Hint: The ratios $A:B$ and $B:C$ are known to us. Make the value of $B$ in both the ratios same and then compare both the ratios.

Complete step-by-step answer:

According to the question, \[A:B = 2:3\] and \[B:C = 4:7\].
If we multiply the first ratio by 4 and second ratio by 3, we’ll get:
$
   \Rightarrow A:B = \left( {2:3} \right) \times 4, \\
   \Rightarrow A:B = 8:12 .....(i) \\
   \Rightarrow B:C = \left( {4:7} \right) \times 3, \\
   \Rightarrow B:C = 12:21 .....(ii) \\
$
Now, if we compare ratios $(i)$ and $(ii)$, the value of B in both the ratios is 12. So, we can easily combine both the ratios as:
$
   \Rightarrow A:B = 8:12,B:C = 12:21 \\
   \Rightarrow A:B:C = 8:12:21 \\
$
Therefore the required ratio $A:B:C$ is $8:12:21$.

Note: We can also solve the question by different method:
As we know that $A:B = 2:3$.
Let the value of A is $2x$ and that of B is $3x$.
Now it is given that \[B:C = 4:7\] and we have already taken the value of B as $3x$. So, we have:
$
   \Rightarrow \dfrac{B}{C} = \dfrac{4}{7}, \\
   \Rightarrow \dfrac{{3x}}{C} = \dfrac{4}{7}, \\
   \Rightarrow C = \dfrac{{21}}{4}x \\
$
Thus we have:
$
   \Rightarrow A = 2x,B = 3x{\text{ and }}C = \dfrac{{21}}{4}x, \\
   \Rightarrow A:B:C = 2:3:\dfrac{{21}}{4}, \\
   \Rightarrow A:B:C = 8:12:21 \\
$